Given 3 diophantine equations:
$$x_1y_1+x_2y_2=x_3y_3+x_4y_4$$
and
$$x_1+x_2 = x_3+x_4$$ and
$$y_1+y_2 = y_3+y_4$$
We're interested in solutions to this system of equations when all variables are positive. I conjecture that any solutions have $x_1=x_3$ or $x_1=x_4$ and $y_1=y_3$ or $y_1=y_4$. Any tips on how to go about proving this? Thanks.